Arithmetic properties of lacunary sums of binomial coefficients

Arithmetic properties of lacunary sums of binomial coefficients Tamás Mathematics Department Occidental College 29th Journées Arithmétiques JA2015, July 6-10, 2015 Arithmetic properties of lacunary sums of binomial coeffs. p1 Outline 1 Notations Previous Work Applications 2 Preparation: Congruences and Div. Properties of Binomial Coeffs 3 Arithmetic properties of lacunary sums of binomial coeffs. p2

Notations Previous Work Applications Notation: p-adic order: ν p (n/k; generating functions Notation n and k positive integers, p prime ν p (k: highest power of p dividing k ν p (n/k = ν p (n ν p (k U n = U(n; m, j, i = ( mn+j i with corresponding generating function l(x = l(x; m, j, i = k=0 U kx k = ( mk+j k=0 i x k, consider the sum T n = T (n; m, j, i = n ( mk+j k=0 i and the generating function of the sequence {T n } n 0 [typically, 0 j < m]: f (x = f (x; m, j, i = n=0 T (n; m, j, ix n. Arithmetic properties of lacunary sums of binomial coeffs. p3 Notations Previous Work Applications Previous Work Previous Work Takács (1978: L. Takács, A sum of binomial coefficients, Mathematics of Computation 32(1978, 1271 1273 Cormier and Eggleton (1976: R.J. Cormier and R.B. Eggleton, Counting by correspondence, Mathematics Magazine 49(1976, 181 186 (2007: T., Asymptotics for lacunary sums of binomial coefficients and a card problem with ranks, Journal of Integer Sequences 10(2007, Article 07.7.2 Sun (2002, and Tollisen and (2004: other definition of lacunary sums of binomial coefficients Arithmetic properties of lacunary sums of binomial coeffs. p4

Notations Previous Work Applications Applications: Asymptotics Theorem 1 (Theorem 3, L (2007. Let i, m, and n be positive integers and j be an integer so that 0 j < min{i + 1, m}. We have that n ( mk + j 1 ( ( m(n + 1 + j i + 1 1 i m i + 1 2(n + 1 + 1 m k=0 for 1 i < n 1/4 as n. ( 1 1 m Arithmetic properties of lacunary sums of binomial coeffs. p5 Notations Previous Work Applications This result with j = 0 was the main tool to answer a question regarding card ranks in an asymptotic sense: pick the first i cards from the top of a standard deck and check the ranks of these cards as they show up kings have the highest rank, 13, while the aces have the lowest, 1; generalized case: the deck of cards has n denominations (of face values 1, 2,..., n in each of m suits; thus, n = 13 and m = 4 for the standard deck p i =P(first card has the uniquely highest rank among the top ( mn i cards of the card deck =??? ANSWER :T (n; m, j, i/. i n ( mk + j 1 ( ( ( m(n + 1 + j i + 1 1 i m i + 1 2(n + 1 + 1 m 1 1 m k=0 for 1 i < n 1/4 as n. Arithmetic properties of lacunary sums of binomial coeffs. p6

Notations Previous Work Applications Theorem 2 (Theorem 1, L (2007. Let i, m, and n be positive integers and p i as defined above. Then p i 1 1 ( 1 1, i 2n m for 2 i n 1/4 as n. Arithmetic properties of lacunary sums of binomial coeffs. p7 Notations Previous Work Applications Applications: Counting Triangles I Another application concerns counting triangles in triangles, cf. Cormier and Eggleton (1976: The total number of triangles in an equilateral triangle of side n tiled by equilateral triangles of side one is 1 ( 16 4n 3 + 10n 2 + 4n 1 + ( 1 n which can be confirmed by observing that { T ( n/2 ; 2, 0, 2, if n is even, T (n + 1; 1, 0, 2 + T ( n/2 ; 2, 1, 2, if n is odd counts the included triangles. Indeed, if the base triangle is pointing up then the first and second terms count the triangles pointing up and down, respectively. Arithmetic properties of lacunary sums of binomial coeffs. p8

Notations Previous Work Applications Applications: Counting Triangles II Figure: from The Book of Numbers by Conway and Guy (1996 n 1 2 3 4 5 6 7 S n = #( + n 1 5 13 27 48 78 118 Example. For n = 5 we have # 5 = 15 + 10 + 6 + 3 + 1 = 35 of size 1,2,3,4, and 5 triangles pointing up and # 5 = 10 + 3 = = 13 of size 1 and 2 triangles pointing down, for a total of S 5 = 48 triangles. BTW, S n satisfies a non-homogeneous linear recurrence of order 3 with characteristic roots -1,1, and 1. Arithmetic properties of lacunary sums of binomial coeffs. p9 Notations Previous Work Applications Applications: Counting Triangles III Arithmetic properties of lacunary sums of binomial coeffs. p10

Plan We use of generating functions, recurrence relations and some congruential and divisibility properties of the binomial coefficients to derive arithmetic properties of {T n } n 0. Arithmetic properties of lacunary sums of binomial coeffs. p11 II Lemma 3 In the case of m = 1 and j = 0 we have that ( n + 1 T n = T (n; 1, 0, i = i + 1 and (1 f (x; 1, 0, i = x i /(1 x i+2. (2 Thus, the following homogeneous linear recurrence holds for T n : main recurrence i+2 ( i + 2 ( 1 s T n s = 0, n i + 2. (3 s s=0 (Question: BTW, do m and j matter? Arithmetic properties of lacunary sums of binomial coeffs. p12

III: The General Case Theorem 4 Let n, m, i, and j, j < m be nonnegative integers. We have that [1] f (x; m, j, i = p(x where p(x = p(x; m, j, i is a (1 x i+2 polynomial of degree i with integer coefficients and the homogeneous linear recurrence (3 holds for T n = T (n; m, j, i with n i + 2 [2] T n = T (n; m, j, i is a polynomial of degree i + 1 in variable n with rational coefficients [3] the leading coefficient of the polynomial for T n is m i /(i + 1!. Arithmetic properties of lacunary sums of binomial coeffs. p13 IV Remark 5 Surprisingly, the main recurrence rel. (3 does not depend on j and m. explore the relation between l(x and f (x alternative derivation of the generating function of the sequence {T n } n 0 : l(x 1 x = n=0 k=0 n ( mk + j x n = i T n x n = f (x (4 n=0 Arithmetic properties of lacunary sums of binomial coeffs. p14

V Theorem 6 Let n, m, i, and j, j < m be nonnegative integers. The homogeneous linear recurrence i+1 s=0 ( i + 1 s ( 1 s U n s = 0, n i + 1 (5 holds for U n = U(n; m, j, i with n i + 1. We have that l(x; m, j, i = r(x (1 x i+1 (6 where r(x = r(x; m, j, i is a polynomial of degree i with integer coefficients. Arithmetic properties of lacunary sums of binomial coeffs. p15 Congruences for some T n I Recurrence relation = Congruences Theorem 7 We have that T (n; p, 0, i 0 mod p if i 0 mod p. Proof. for all terms in the definition of T n : ( kp i 0 mod p if i 0 mod p by carry counting according to the Kummer s Theorem The case with i 0 mod p is discussed in Theorem 8, and its proof requires some preparations. In fact, we need Lucas Theorem and the Jacobstahl Kazandzidis congruences rather than the recurrence relation (except for part (c of Theorem 8. Arithmetic properties of lacunary sums of binomial coeffs. p16

Congruences for some T n II Theorem 8 For 0 j < p and t 0 integers we have that ( n + 1 T n = T (n; p, j, tp mod p. t + 1 In addition, we also have that T n = T (n; p, 0, tp ( n + 1 mod p q t + 1 with q = { 2, if p = 2 or 3, 3, if p 5. Arithmetic properties of lacunary sums of binomial coeffs. p17 Congruences for some T n III Furthermore, if p = 2 and t 0 is even then T n = T (n; 2, 0, 2t ( n + 1 mod 2 ν2(t+1. t + 1 Arithmetic properties of lacunary sums of binomial coeffs. p18

p-adic order of some T n I Now we turn to the question about the magnitude of ν p (T n. Definition of Polynomial Divisibility Let a(x and b(x be polynomials in variable x. We say that b(x divides a(x and write that a(x 0 mod b(x if there exists a polynomial c(x so that a(x = c(xb(x. For instance, when we say that n or n + 1 divides T (n; m, j, i we mean the divisibility of the corresponding polynomials. Theorem 9 For i 0 we have that n + 1 divides T (n; m, j, i. Arithmetic properties of lacunary sums of binomial coeffs. p19 p-adic order of some T n II By setting n = p N 1 with a large N, Theorem 9 implies Corollary 10 The p-adic order ν p (T (n; m, j, i can be arbitrarily large for any prime p. Arithmetic properties of lacunary sums of binomial coeffs. p20

p-adic order of some T n III Numerical experiments suggest the following improvement. Although, at first, it seems difficult to prove that n ( mk + j k=0 i is divisible by n, the previous theorem comes to the rescue. Theorem 11 For 0 j < i we have that n(n + 1 divides T (n; m, j, i. Arithmetic properties of lacunary sums of binomial coeffs. p21 p-adic order of some T n IV It is possible, however, that a subsequence of {ν p (T n } n 0 remains constant. A case in point is parts (a and (b of Theorem 12. Theorem 12 Let p be a prime. We set n 0 = n 0 (m, j, i = max{0, i j m }, T s = T (s; m, j, i and with M = ν p (T s. argmin ν p (T (s; m, j, i = s (7 n 0 s n 0 +i+1 (a (special case If i = p 2 then there is an arithmetic sequence {s + Kp} K 0 with some integer s n 0 so that the subsequence {ν p (T s +Kp} K 0 of the original sequence {ν p (T n } n 0 remains Arithmetic properties of lacunary sums of binomial coeffs. p22

p-adic order of some T n V constant. We set s by (7 and have, with T s +Kp = T (s + Kp; m, j, p 2, that for all nonnegative integers K. ν p (T s +Kp = M (b If i 0, and the minimum in (7 is uniquely taken at s, i.e., M = ν p (T s < ν p (T s for all s : s s, n 0 s n 0 + i + 1, then ν p (T s +K(i+2 = M holds with T s +K(i+2 = T (s + K(i + 2; m, j, i. (c In general, we have the universal lower bound M ν p (T n for all n s. Arithmetic properties of lacunary sums of binomial coeffs. p23 Example 13 (with p = 5 and i = p 2 = 3 we can easily see that for n 1 ν 5 (T (n; 5, 2, 3 = ν 5 ( 5 24 n(n + 1(5n + 2(5n + 7 ν 5 (n + 1, if n 0 mod 5, = ν 5 (n + 1 + 1, if n 4 mod 5, 1, otherwise; minimum mod 5 and 25 periods are π(5 = 1 and π(5 2 = 5 {T n } n 0 = {0, 35, 255, 935, 2475, 5400,... } and {ν 5 (T n } n=5 n=1 = {1, 1, 1, 2, 2}; ν 5(T 5 n 1 = ν 5 (T 5 n = n + 1 if n 1 and ν 5 (T s +5n = 1 if n 0 and s = 1, 2, 3 T n : poly. of degree 4 w/leading coeff. 5 3 /4!; n(n + 1 T n Arithmetic properties of lacunary sums of binomial coeffs. p24

p-adic order of some T n VIII Remark 14 part (b of Theorem 12 does not seem to have practical value as we found only a few sets of parameters when its conditions are satisfied and these cases are also covered by part (a; on the other hand, sometimes it is easy to see why the p-adic order remains constant without invoking Theorem 12; indeed, numerical experiments suggest that for small values of i we often encounter the case where ν p (T n = Aν p (n + Bν p (n + 1 + C with some positive rational constants A, B, and C; then ν p (T n = C provided that n 0 or 1 mod p, see Example 13 Arithmetic properties of lacunary sums of binomial coeffs. p25 Congruences and Divisibility Properties of Binomial Coefficients I Three theorems comprise the most basic facts re: divisibility and congruence properties of binomial coefficients. (Assume that 0 k n. Theorem 15 (Kummer, 1852. The power of a prime p that divides the binomial coefficient ( n k is given by the number of carries when we add k and n k in base p. Arithmetic properties of lacunary sums of binomial coeffs. p26

Congruences and Divisibility Properties of Binomial Coefficients II From now on M and N will denote integers such that 0 M N. Theorem 16 (Lucas, 1877. Let N = (n d,..., n 1, n 0 p = n 0 + n 1 p + + n d p d and M = m 0 + m 1 p + + m d p d with 0 n i, m i p 1 for each i, be the base p representations of N and M, respectively. ( ( ( ( N n0 n1 nd mod p. M m 0 m 1 m d Arithmetic properties of lacunary sums of binomial coeffs. p27 Congruences and Divisibility Properties of Binomial Coefficients III We also use the following generalization of the Jacobstahl Kazandzidis congruences. Theorem 17 (Corollary 11.6.22 in Cohen (2007. Let M and N such that 0 M N and p prime. We set R = p 4 NM(N M ( N M and have ( ( pn pm 1 B p 3 3 p3 NM(N M (1 + 45NM(N M ( N M ( 1 M(N M P(N, M ( N M (N M modr, if p 5, modr, if p = 3, modr, if p = 2, Arithmetic properties of lacunary sums of binomial coeffs. p28

Congruences and Divisibility Properties of Binomial Coefficients IV where P(N, M = 1+6NM(N M 4NM(N M(N 2 NM +M 2 +2(NM(N M 2. Arithmetic properties of lacunary sums of binomial coeffs. p29 for the I use generating functions to derive and prove recurrence relations that are interesting on their own Arithmetic properties of lacunary sums of binomial coeffs. p30

for the II Proof of Lemma 3. Id. (1 and (2 are obvious by the so called hockey-stick id. ( ( ( ( i i + 1 n n + 1 + + + = i i i i + 1 and f (x; 1, 0, i = T n x n = n=0 n=i ( n + 1 x n = x i i + 1 n=0 ( n + i + 1 ( (i + 2 = x i ( x n = x i /(1 x i+2. n n=0 The denominator guarantees the main recurrence (3. i + 1 x n (8 Arithmetic properties of lacunary sums of binomial coeffs. p31 I We prove Theorem 6 before Theorem 4. Proof of Theorem 6. Now we show that l(x = q(x/(1 x i+1 where q(x = q(x; m, j, i is a polynomial of degree i with integer coefficients. For example, with m = 1 and j = 0 we get that l(x = x i /(1 x i+1 in a similar fashion to (8. For all m 1, we prove by induction on i 0 that the denominator of l(x = l(x; m, j, i in (6 is (1 x i+1 independently of j while the numerator is a polynomial of degree i with integer coefficients. statement is obviously true for i = 0 since l(x; m, j, 0 = 1/(1 x assume that it is also true 0, 1, 2..., i 1 and prove it for i Arithmetic properties of lacunary sums of binomial coeffs. p32

More proofs A/ I To this end, we observe that ( ( ( ms + j m(s 1 + j m(s 1 + m + j 1 = i i i 1 ( m(s 1 + j 2 + +... i 1 ( m(s 1 + j + i 1 by the hockey-stick identity. Arithmetic properties of lacunary sums of binomial coeffs. p33 More proofs A/ II multiply both sides by x s and sum up the terms: (( ( ms + j m(s 1 + j (1 xl(x; m, j, i = i i = s=1 j 1 s=0 l(x; m, s, i 1 + x m 1 s=j for j 1 and (1 xl(x; m, 0, i = x m 1 s=0 l(x; m, s, i 1 for j = 0 divide both sides by 1 x x s l(x; m, s, i 1 The degree of the polynomial in the numerator also follows. The form of the generating function l(x guarantees the validity of the recurrence (5. Arithmetic properties of lacunary sums of binomial coeffs. p34

More proofs A/ III Proof of Theorem 4. part (1: use Theorem 6 and identity (4. The form of the generating function f (x guarantees the validity of the main recurrence (3 parts (2 and (3: use a bivariate generating function g(x, y in which y marks the summation variable k. g(x, y = = mn+j i=0 n k=0 n ( mk + j x i y k = i k=0 n k=0 mk+j y k i=0 ( mk + j i x i y k (1 + x mk+j = (1 + x j 1 (y(1 + xm n+1 1 y(1 + x m. Arithmetic properties of lacunary sums of binomial coeffs. p35 More proofs A/ IV With y = 1 we derive the generating function h(x = g(x, 1 = mn+j i=0 = (1 + xm(n+1+j (1 + x j (1 + x m 1 T (n; m, j, ix i = (1 + x j (1 + xm(n+1 1 (1 + x m 1. Arithmetic properties of lacunary sums of binomial coeffs. p36

More proofs A/ V Since (1 + x m 1 = mx it follows that ( 1 + m 1 2 x + (m 1(m 2 6 x 2 +..., T (n; m, j, i = [x i ] h(x i+1 (( 1 m(n + 1 + j = m s s=0 [x i+1 s ] 1 ( j (9 s 1 + m 1 2 x + (m 1(m 2 6 x 2 +... where the denominator in the last expression does not depend on n = T n is a polynomial in n of degree i + 1 with rational coefficients and the leading coefficient, i.e., that of n i+1, is m i /(i + 1! Arithmetic properties of lacunary sums of binomial coeffs. p37 More proofs A/ VI Proof of Theorem 8. follows by Lucas Theorem and the Jacobstahl Kazandzidis congruence, for 0 j < p we get that ( ( kp+j tp k ( j ( t 0 k t mod p by Lucas s Theorem and T n = T (n; p, j, tp n ( k ( k=0 t = n+1 t+1 mod p by the hockey-stick identity for p 3 we have ( ( kp tp k t mod p q by the Jacobstahl Kazandzidis congruence = T n = T (n; p, 0, tp n ( k ( k=0 t = n+1 t+1 mod p q. if p = 2 and t is even then we can use the 2-exponent ν 2 (t + 1 by the same reasoning and utilizing the factor t in the modulus Arithmetic properties of lacunary sums of binomial coeffs. p38

More proofs A/ VII if p = 2 and t is odd then the terms of T n satisfy the congruence ( ( 2k k ( 1 t(k t mod 4, 2t t which is ( ( k t if k is odd and k t 0 or 2 mod 4 and thus, ( ( k t k t mod 4 if k is even the statement follows Arithmetic properties of lacunary sums of binomial coeffs. p39 More proofs A/ VIII Proof of Theorem 9. obvious if i = 0 otherwise, the statement follows by identity (9 and some congruences since ( ( m(n + 1 + j j 0 mod (n + 1; (10 s s 1 in fact, if i < j then for any s, 0 s s i + 1 j we have m(n + 1 + j s j s mod (n + 1 which yields (10 2 if s j i then the above argument also applies 3 if j < s i + 1 then (11 implies (10 again by ( ( m(n + 1 + j j(j 1... (j s + 1 j 0 mod (n+1. (11 s s! s Arithmetic properties of lacunary sums of binomial coeffs. p40

More proofs B/ Proof of Theorem 11. Theorem 9 takes care of divisibility by n + 1. For the other part, we divide the sum into two parts ( mn + j T (n; m, j, i = T (n 1; m, j, i +. i 1 first term is divisible by n by Theorem 9 2 second term: we proceed similarly to congruence (11: ( mn + j i j(j 1... (j i + 1 i! since 0 j < i, and the proof is complete 0 mod n Arithmetic properties of lacunary sums of binomial coeffs. p41 More proofs C/ Proof of Theorem 12. ( part (a: we assume that p is a prime and i = p 2 = i+2 s 0 mod p for 1 s i + 1 and the main recurrence (3 taken mod p guarantees that M = ν p (T s = ν p (T s +i+2 = ν p (T s +2(i+2 =.... part (b: the mod p M+1 version of the main recurrence (3 yields the result part (c: mathematical induction on n n 0 (m, j, i and (3 Arithmetic properties of lacunary sums of binomial coeffs. p42

for i 0 we have that n + 1 divides T (n; m, j, i = n ( mk+j k=0 i (and for 0 j < i we have that n(n + 1 divides T (n; m, j, i = n k=0 ( mk+j i thus, the p-adic order ν p (T (n; m; j, i can be arbitrarily large for any prime p some subsequence of {ν p (T n } n 0 might remain constant, e.g., if i = p 2 then the arithmetic subsequence {s + Kp} K 0 of the indices with a proper choice of s results in ν p (T s +Kp = M; also, by Theorem 8 we get that ν p (T (Kp 2 + p 1, p, j, p(p 1 = 0 for K 0 and 0 j < p Arithmetic properties of lacunary sums of binomial coeffs. p43 Further Study Find higher order congruences for T n in Theorem 8 and in general (when i is not of form tp. Better understand the factors of T n /(n(n + 1 (cf. Theorem 11 and Example 13. Use p-sections in the main recurrence to improve the exponent of the prime p in congruences for T n. Arithmetic properties of lacunary sums of binomial coeffs. p44

The p-sected modular version of recurrence (3 We obtain the p-sected modular version of the main recurrence (3 in the following theorem. We note that p-sections might help in uncovering some inconspicuous arithmetic properties. Theorem 18 With n (t + 1p and t 0, we have for T n = T (n; p, 0, (t + 1p 2 that t+1 s=0 ( (t + 1p sp ( 1 sp T n sp 0 mod p. (12 Remark 19 Note that typically the exponent of p can be increased because of part (c of Theorem 12. Arithmetic properties of lacunary sums of binomial coeffs. p45 More proofs D/ Proof of Theorem 18. We apply the main recurrence (3. Since now i + 2 = (t + 1p, we get that ( (t + 1p 0 mod p l if l 0 mod p by Lucas Theorem. The remaining terms yield (12. Arithmetic properties of lacunary sums of binomial coeffs. p46

I H. Cohen, p-adic Γ and L functions, lecture notes, Barcelona, 2005 Number Theory, vol 2: Analytic and Modern Tools, Graduate Texts in Mathematics, Springer, 2007. L. Comtet, Advanced Combinatorics. The art of finite and infinite expansions. D. Reidel, Publishing Co., Dordrecht, enlarged edition, 1974. J.H. Conway and R.K. Guy, The Book of Numbers. Springer, New York, 1996. R.J. Cormier and R.B. Eggleton, Counting by correspondence, Mathematics Magazine 49(1976, 181 186. Arithmetic properties of lacunary sums of binomial coeffs. p47 II F.T. Howard and R. Witt, Lacunary sums of binomial coefficients. Applications of Fibonacci numbers, Vol. 7 (Graz, 1996, 185-195, Kluwer Acad. Publ., Dordrecht, 1998. Y.H.H. Kwong, Periodicities of a class of infinite integer sequences modulo M, J. Number Theory 31(1989, 64-79. T., Asymptotics for lacunary sums of binomial coefficients and a card problem with ranks, Journal of Integer Sequences 10(2007, Article 07.7.2. L. Takács, A sum of binomial coefficients, Mathematics of Computation 32(1978, 1271 1273. Arithmetic properties of lacunary sums of binomial coeffs. p48

III G. Tollisen and T., A congruential identity and the 2-adic order of lacunary sums of binomial coefficients, INTEGERS, Electronic Journal of Combinatorial Number Theory, A4(2004, 1 8. Z.-W. Sun, On the sum k r (mod m ( n k and related congruences, Israel J. Math. 128(2002, 135 156. Arithmetic properties of lacunary sums of binomial coeffs. p49